The closed-form solutions for \(\mathcal {I}\left(x,\theta \right)\)  are derived in terms of the confluent Lauricella's hypergeometric function, which is defined as [1]}, [2]}
\(\Phi _1^{(n)}(m;p_1, p_2\ldots p_{n-1};q;z_1,z_2,\ldots z_n)&=\frac{\Gamma (q)}{\Gamma (m)\Gamma (q-m)}\int _0^1v^{m-1}(1-v)^{q-m-1}\nonumber \\& \times \prod _{i=1}^{n-1}(1-v\,z_i)^{-p_i}\,\exp (v\,z_n)dv,\) 
